Our goal is to develop a comprehensive framework for understanding synchronization in circuits containing biological neurons, where synchronization is broadly defined to include all phase-locked rhythmic activity. We will focus on two types of biological exemplars of neural oscillators: repetitively spiking neurons with a gradual dependence of the firing frequency on stimulus current, and bursting neurons that apparently have a single slow variable. A large number of neural oscillators are likely to fall into one of these two categories, hence these results should be general and useful in order to compare and contrast phase resetting and phase locking in single spike firing versus bursting neurons. The experimental preparations were chosen because they contain spiking or bursting neurons that are easily identified, readily isolated, and oscillate with minimal variability. This makes them an optimal in vitro proving ground for answering the following questions, using theoretical methods based on phase resetting curves (PRCs): Are PRCs sufficient to predict phase locking and convergence in pairs of coupled spiking neurons? Can PROs predict the existence and stability of m:n phase-locking modes? Can the PRC in response to excitatory stimuli be predicted from the voltage trajectory of a bursting neuron? Can we predict the activity of pairs of bursting neurons from their PROs for excitatory coupling and for variable burst durations? The answers should have wide applicability in the study of central pattern generators that produce repetitive motor activity, as well as to the collective synchronization phenomena underlying various aspects of cognition. Epileptic seizures are associated with excessive synchronization in certain brain regions, as is the tremor associated with Parkinson's disease. A better understanding of the general mechanisms of synchronization may eventually suggest new therapeutic approaches for these diseases. [unreadable] [unreadable]